Find necessary and sufficient conditions on A so that $e^{At}$ is bounded for all $t>0$

Question

Let A be an n × n real matrix. Find necessary and sufficient conditions on A so that $e^{At}$ is bounded for all $t>0$.

I am trying to figure this out. Can anyone give me a hint? Does it have to do with the eigenvalues of A?

Answer 1

Yes. It has to do with the eigenvalues of $A$. Start by answering the question when $A$ is diagonalizable over $\Bbb C$. In the general case, Jordan canonical form will help.